Leximin is a common approach to multi-objective optimization, frequently employed in fair division applications. In leximin optimization, one first aims to maximize the smallest objective value; subject to this, one maximizes the second-smallest objective; and so on. Often, even the single-objective problem of maximizing the smallest value cannot be solved accurately. What can we hope to accomplish for leximin optimization in this situation? Recently, Henzinger et al. (2022) defined a notion of \emph{approximate} leximin optimality. Their definition, however, considers only an additive approximation. In this work, we first define the notion of approximate leximin optimality, allowing both multiplicative and additive errors. We then show how to compute, in polynomial time, such an approximate leximin solution, using an oracle that finds an approximation to a single-objective problem. The approximation factors of the algorithms are closely related: an $(\alpha,\epsilon)$-approximation for the single-objective problem (where $\alpha \in (0,1]$ and $\epsilon \geq 0$ are the multiplicative and additive factors respectively) translates into an $\left(\frac{\alpha^2}{1-\alpha + \alpha^2}, \frac{\epsilon}{1-\alpha +\alpha^2}\right)$-approximation for the multi-objective leximin problem, regardless of the number of objectives. Finally, we apply our algorithm to obtain an approximate leximin solution for the problem of \emph{stochastic allocations of indivisible goods}. For this problem, assuming sub-modular objectives functions, the single-objective egalitarian welfare can be approximated, with only a multiplicative error, to an optimal $1-\frac{1}{e}\approx 0.632$ factor w.h.p. We show how to extend the approximation to leximin, over all the objective functions, to a multiplicative factor of $\frac{(e-1)^2}{e^2-e+1} \approx 0.52$ w.h.p or $\frac{1}{3}$ deterministically.

翻译：Leximin是多目标优化中一种常见方法，常用于公平分配问题。在leximin优化中，首先最大化最小的目标值；在此基础上，最大化第二小的目标值；依此类推。然而，即使是最大化最小值的单目标问题也通常难以精确求解。在此情况下，我们能对leximin优化实现何种目标？最近，Henzinger等人（2022）定义了近似leximin最优性的概念，但仅考虑了加法近似。本文首先定义允许乘法误差和加法误差的近似leximin最优性概念，随后展示如何利用求解单目标问题近似的预言机，在多项式时间内计算出此类近似leximin解。这些算法的近似因子密切相关：若单目标问题存在$(\alpha,\epsilon)$-近似（其中$\alpha\in(0,1]$和$\epsilon\geq0$分别为乘法因子和加法因子），则无论目标数量多少，该近似可转化为多目标leximin问题的$\left(\frac{\alpha^2}{1-\alpha+\alpha^2},\frac{\epsilon}{1-\alpha+\alpha^2}\right)$-近似。最后，我们将算法应用于不可分割物品的随机分配问题，以获得近似leximin解。对于该问题，假设子模目标函数，单目标平均主义福利可仅以乘法误差高概率实现最优近似因子$1-\frac{1}{e}\approx 0.632$。我们进一步将该近似扩展到所有目标函数的leximin，使其乘法因子高概率达到$\frac{(e-1)^2}{e^2-e+1}\approx 0.52$，或确定性地达到$\frac{1}{3}$。